Equivariant chromatic and motivic homotopy theory pdf

The new model structure introduced here samples a comparison to the one by voevodsky and hukrizormsby. This includes the naive gspectra which constitute the actual stabilization of equivariant homotopy theory, but is more general, one speaks of genuine g g spectra. The workshop will bring together experts working on the abstract foundations as well as concrete computational applications. Homotopy theory covers a wideswath of algebraic topology, exploring every. Chromatic homotopy theory is the study of stable homotopy theory and specifically of. Wed hope to use chromatic methods to understand the equivariant theory. Theres a certain analogy between this and chromatic homotopy theory just as in chromatic homotopy theory a spectrum is built from its knlocalizations, here a spectrum is built from its geometric xed point spectra for the various subgroups of g. E ect of a1localization on ring and module structures and on modlcompletions 5.

Equivariant and motivic homotopy theory isaac newton. The electronic computational homotopy theory seminar is an international research seminar on the topic of computational homotopy theory. We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks x g, where g is a linearly reductive linear algebraic group. Andrew blumberg, equivariant homotopy theory, 2017 pdf, github. Here g is a nite group that acts on a small category i. Rigidity in equivariant stable homotopy theory irakli patchkoria for any nite group g, we show that the 2local gequivariant stable homotopy category, indexed on a complete guniverse, has a unique equivariant. In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom diecks splitting theorem for the fixed points of a suspension spectrum. These notes are an overview of equivariant stable homotopy theory.

Thick ideals in equivariant and motivic stable homotopy. Motivic homotopy theory is a homotopy theory of schemes in which the a. Gunnar carlsson, roy joshua, equivariant motivic homotopy theory, arxiv. In equivariant stable homotopy theory for a nite group g, unpublished work of strickland str10, see chapter 3, contains a partial classi cation of thick ideals in the category shg f. I the homotopy limit problem for karoubis hermitian ktheory 23 was posed by thomason in 1983 43. Glossary for stable and chromatic honotopy theory pdf. Mathematisches forschungsinstitut oberwolfach homotopy theory. A picture of the corpse of equivariant homotopy theory crawling out of a grave.

The construction uses the idea of equivariant k theory of automorphisms to produce a tower for equivariant k theory, where the successive layers are weak. The book begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the sullivan conjecture that emphasizes its relationship with classical smith theory. Equivariant, chromatic, and motivic homotopy theory, northwestern. Then we have axiomatic modeltheoretic homotopy theory, stable homotopy theory, chromatic homotopy theory. The balmer spectrum of the equivariant homotopy category. The new model structure introduced here will be compared to those by voevodsky and hukrizormsby. Hopkins 8, stable equivariant homotopy theory controls the chromatic decomposition of stable homotopy theory. We extend to this equivariant setting the main foundational results of motivic homotopy theory. For example, we have simplicial homotopy theory, where one studies simplicial sets instead of topological spaces.

An important application of equivariant coarse homotopy theory is in the study of assembly. Equivariant stable homotopy theory over some topological group g g is the stable homotopy theory of gspectra. It then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant. The main reference for this theory is the ams memoir 16 by mandell and may. This conference will focus on the interplay and parallelisms between equivariant and motivic homotopy theory, providing a forum in which researchers can share insights and techniques from both disciplines. In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. All notes links are to pdf files, and they are coded by the initials of the notetaker. My talk the coefficients of stable equivariant cobordism for a finite abelian group on joint work with will abram. There will be a conference on topological field theory and related geometry, topology, and category theory to be held at northwestern university during the week of memorial day may 2529, 2009. What is modern algebraic topologyhomotopy theory about. Conference on equivariant, chromatic and motivic homotopy theory, northwestern university 20. Motivic homotopy theory was introduced by voevodsky voe98. A number of authors have studied links between g equivariant and motivic stable homotopy theory, including myself and jeremiah heller in a paper that was recently accepted to the transactions of the ams, galois equivariance and stable motivic homotopy theory, previously written about on this blog here.

Nonnilpotent selfmaps in motivic and equivariant homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and k theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations. Michael andrews mit aravind asok university of southern california bert guillou university of kentucky. Equivariant and motivic homotopy are two of the most active areas in algebraic topology, bringing together researchers in topology, algebra, representation theory, and algebraic geometry. In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action instead of a space with group action, as in stable homotopy theory. September 24, 2018 6\picez4 and tools to compute it equivariant and motivic homotopy. The second author was supported by the ihes, the mpi bonn and a grant from the nsa. Relations to other parts of chrok and some intriguing question chromatic homotopy theory. Galois equivariance and stable motivic homotopy theory. This was vastly generalized and studied more thoroughly in. A survey of equivariant stable homotopy theory gunnar carlsson receiced 15 march 1991. Following this line of thought, an entire stable homotopy category can be created. Ams sectional meeting, special session of computational algebraic.

Algebraic structures in equivariant homotopy theory hood chatham juvitop february 22, 2016 g equivariant abelian groups lukas told us last time that a weak homotopy equivalence in the category of gspectra is a map f. A proposal for the establishment of a dfgpriority program. As far as i understand, simplicial techniques are indispensible in modern topology. The canonical homomorphisms of topological g spaces are g equivariant continuous functions, and the canonical choice. Equivariant and motivic homotopy theory reed college, may 3031, 2015. At the workshop on motivic and algebrogeometric homotopy theory i gave two lectures about galois equivariant motivic phenomena in arithmetic. As a relatively new field of research this subject has quickly turned into a wellestablished area. This repository holds lecture notes for andrew blumbergs class on equivariant homotopy theory at ut austin in spring 2017. In this paper, following graysons approach, we establish an equivariant motivic spectral sequence for a ne noetherian regular gschemes called the equivariant grayson spectral sequence.

Equivariant homotopy theory is homotopy theory for the case that a group g acts on all the topological spaces or other objects involved, hence the homotopy theory of topological gspaces. We will show how to use galois descent methods to compute some new picard groups arising in chromatic homotopy theory. He was motivated by his work with atiyah 9 on equivariant k theory, generalizing an. We show that it allows to detect equivariant motivic weak equivalences on fixed points and how this property leads to a. Topics include any part of homotopy theory that has a computational flavor, including but not limited to stable homotopy theory, unstable homotopy theory, chromatic homotopy theory, equivariant homotopy theory, motivic homotopy theory, and k theory. Pa1 chromatic and motivic aspects of stable homotopy theory. This leads to a theory of motivic spheres s p,q with two indices.

N2 for a finite galois extension of fields lk with galois group g, we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence. This category has many nice properties which are not present in the unstable homotopy category of spaces, following from the fact that the suspension functor becomes. Northwestern workshop on equivariant, chromatic, and motivic homotopy theory, march 20 19. The picard group of the stable homotopy category is known to contain only suspensions of the sphere spectrum. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from. Equivariant stable homotopy theory 5 isotropy groups and universal spaces. Peter may, equivariant homotopy and cohomology theory, cbms regional conference series in mathematics, vol. Motivic homotopy theory is a homotopy theory of schemes in which the.

Equivariant motivic homotopy theory 3 asoks program aso, 2. Dec 01, 20 in this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. Equivariant coarse homotopy theory and coarse algebraic k. Shg, which is the full subcategory of compact objects in the g equivariant stable homotopy category. As we explain below, this question is the equivariant analogue of the celebrated chromatic ltration in classical stable homotopy theory, due to devinatzhopkinssmith dhs88,hs98. Theres a certain analogy between this and chromatic homotopy theory just as in chromatic homotopy theory a spectrum is built from its knlocalizations, here a spectrum is built from. While our main interest is the case when the group is pro nite, we discuss our results in a more general setting so. After certain localizations, however, the picard group becomes much more interesting. Schwede in global equivariant homotopy theory, the role of complex bordism is the universal globally complex oriented theory. By the construction of complex oriented cohomology theories from formal groups via the. Picard groups and to the interaction between chromatic and equivariant stable homotopy theory.

Equivariant homotopy theory in problems in homotopy theory. Algebraic topology summer graduate school, msri, june 20 intro to stable homotopy and spectra i, ii, iii, and iv videos for four 75 minute lectures. We compare the motivic slice filtration of a motivic spectrum over spec k with the c 2 equivariant slice filtration of its equivariant betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. The field has become more active recently because of its connection to algebraic k theory. The six operations in equivariant motivic homotopy theory. I the homotopy limit problem for karoubis hermitian k theory 25 was posed by thomason in 1983 48. Important work on this topic has been done by strickland str12. A picture of a viking longboat moving motivic homotopy theorists. Via the work of devinatz and hopkins 8, stable equivariant homotopy theory controls the chromatic decomposition of stable homotopy theory. Rigidity and algebraic models for rational equivariant stable homotopy theory. The basic framework of equivariant motivic homotopy theory. I the homotopy limit problem for karoubis hermitian ktheory 25 was posed by thomason in 1983 48. The slice filtration in equivariant or motivic homotopy theory is an analog of.

Descent and chromatic homotopy theory, strasbourg 2019. Equivariant homotopy theory has been a fundamental component of algebraic topology since its inception. It provides a framework for applying techniques from algebraic topology to the study of smooth schemes over a base. On the theory and applications of torsion products with. Mackey functors, km,ns, and roggraded cohomology 25 6. Characters in global equivariant homotopy theory a thesis submi. While our main interest is the case when the group is profinite, we discuss our results in a. We are now in the modern era of stable homotopy theory, with current topics such as topological modular forms and its variants, motivic stable homotopy theory, the study of commutative ring spectra and their localisations, galois.

What is the universal property that the coefficients enjoy. Workshop in equivariant, chromatic, and motivic homotopy theory. Contact information educational background academic. Marc hoyois, the six operations in equivariant motivic homotopy theory, arxiv. Brown representability in equivariant motivic homotopy theory.

The cancellation problem asks for the correctness of the implication x a1. In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complexoriented cohomology theories from the chromatic point of view, which is based on quillens work relating cohomology theories to formal groups. Ams sectional meeting, special session of computational algebraic topology, university of akron, october 2012 20. In the modern treatment of stable homotopy theory, spaces are typically replaced by spectra. Galois equivariance and stable motivic homotopy theory american.

Equivariant stable homotopy theory with lewis, steinberger, and with contributions by mcclure a brief guide to some addenda and errata pdf american mathematical society memoirs and asterisque at ams memoirs 142. Equivariant, chromatic, and motivic homotopy theory, northwestern university, march 20. I the homotopy limit problem for karoubis hermitian k theory 23 was posed by thomason in 1983 43. The balmer spectrum of the equivariant homotopy category of a. During the period septemberdecember 2002 a research programme entitled new contexts for stable homotopy theory was staged at the isaac newton institute for mathematical research. Contents 1 introduction3 2 chromatic homotopy theory8 3 pdivisible. The second part of the thesis, which consists of one paper, is about the equivariant homotopy theory of socalled gdiagrams. Shg, which is the full subcategory of compact objects in the gequivariant stable homotopy category. In this work, we present an alternative equivariant motivic homotopy theory, based on a slight variation of a nisnevichstyle grothendieck topology on the category of smooth gschemes over a eld. To compute the homotopy groups of motivic spheres would also yield the classical stable homotopy groups of the spheres, so in this respect a 1 homotopy theory is at least as complicated as classical homotopy theory.

Our research aims at formulating and solving groundbreaking problems in motivic homotopy theory. Homotopy harnessing higher structures isaac newton institute. Lectures on equivariant stable homotopy theory contents. While our original interest was in the case of profinite group actions on smooth schemes, we discuss our results in as broad a setting as. As examples of equivariant coarse homology theories we discuss equivariant ordinary coarse homology and equivariant coarse algebraic khomology of an additive category. The six operations in equivariant motivic homotopy theory arxiv.

An algebraic model for rational s1 equivariant stable homotopy. Rigidity in equivariant stable homotopy theory irakli patchkoria for any nite group g, we show that the 2local g equivariant stable homotopy category, indexed on a complete guni. Thick ideals in equivariant and motivic stable homotopy categories. The norm functors in stable equivariant homotopy theory can be constructed in essentially the same. Equivariant motivic homotopy theory gunnar carlsson and roy joshua abstract. A lot of the interesting work in chromatic homotopy theory regarding. Algebraic structures in equivariant homotopy theory. Workshop on group actions in homotopy theory, university of copenhagen, august 20 18. We give a method for computing the c 2 equivariant homotopy groups of the betti realization of a pcomplete cellular motivic spectrum over.